Geodesic finite elements of higher order (Q2794704)

For the Dirichlet problem on the domain \( \Omega \subset \mathbb R^{d}\), reduced to the variational form NEWLINE\[NEWLINE\min J(v)=\int_{\Omega}|\nabla v|^{2}dx \qquad \text{in}\quad H^{1}(\Omega,S^{2}),NEWLINE\]NEWLINE NEWLINE\[NEWLINE v = v_{D} \qquad \text{on} \quad \partial \Omega, NEWLINE\]NEWLINE where \( S^{2}\) is the unit sphere in \( \mathbb R^{d+1}\), the finite element method is applied to approximate solve this problem. The space of finite element functions is constructed using the Lagrangian interpolation with polynomials of the order \(p\) on either element \(T\) of the grid, named the geodesic finite element method.